To overcome this, in [1]}, it was suggested to take into account the (time) evolution of all possible multi-point spatial correlations. Instead of a single probability measure on the phase space \(U\) , such a statistical solution is a family of probability measures on the phase space and products of the phase space \(U^k\) , for \(k\in \mathbb {N}\) , corresponding to the multi-point correlations. Hence, one can interpret the solution as a measure-valued solution augmented with information about higher order spatial correlations.
From a practical point of view, this approach is very natural, as often only averaged quantities of interest of the fluid flow can be observed. Moreover, it is also in line with Kolmogorov's turbulence theory, as this theory studies statistical properties of the fluid and makes predictions about these.
The system of equations that arises for the higher order correlations is also known as the Friedman–Keller infinite chain of moment equations [2]}, [3]} and finite closure relation for this infinite family of equations have been studied for small and large Reynolds numbers in [4]}, [5]}, [6]}, [7]}.
